Given any initial network \({\mathcal {N}}_0\) , Theorem REF  guarantees the existence of a flowout \({\mathcal {N}}(t)\) , i.e., a solution of
the network flow, for some possibly short time-interval \((0,T)\)  which is a regular network for all such positive times. It is quite
conceivable, however, that at some later time the network \({\mathcal {N}}(t)\)  degenerates and becomes irregular at some first
positive time. There are various ways this might happen. For example, some curve in \({\mathcal {N}}(t)\)  might shrink to zero length
at time \(t_0\)  so that the limit of \({\mathcal {N}}(t)\)  as \(r \nearrow t_0\)  is an irregular network in which the vertex which is the limit
of that edge has valence four. Another type of degeneration occurs if \({\mathcal {N}}(t)\)  has a closed region bounded by fewer than
six curves. One can prove that the area of the enclosed region decreases monotonically and shrinks to zero in finite time.
At that time the lengths of some or all of the boundary curves of that region have decreased to zero, so the limiting
network will have a vertex of valence four or five. In such limits, the curvature of the curves around this region is
unbounded above, cf. [1]}. We refer to the well-known numerical simulations of network flow that appear
on Ken Brakke's webpage; these phenomena all can be seen there.
